The following, which illustrates the shopkeeper's paradox, was presented by Lewis Carroll in the July 1894 issue of Mind.

What,

said Uncle Jim. *nothing* to do?Then come along with me down to Allen’s. And you can just take a turn while I get myself shaved.

All right,

said Uncle Joe. And the Cub had better come too, I suppose?

The Cub

was *me*, as the reader will perhaps have guessed for himself. I’m turned *fifteen*—more than three months ago; but there’s no sort of use in mentioning *that* to Uncle Joe; he’d only say go to your cubbicle, little boy!

or Then I suppose you can do cubbic equations?

or some equally vile pun. He asked me yesterday to give him an instance of a Proposition in *A*. And I said All uncles make vile puns

. And I don’t think he liked it. However, that’s neither here nor there. I was glad enough to go. I *do* love hearing those uncles of mine chop logic,

as they call it; and they’re desperate hands at it, *I* can tell you!

That is not a logical inference from my remark,

said Uncle Jim.

Never said it was,

said Uncle Joe: it’s a Reductio ad Absurdum

.

An

chuckled Uncle Jim.
*Illicit Process of the Minor!*

That’s the sort of way they always go on, whenever *I’m* with them. As if there was any fun in calling me a Minor!

After a bit, Uncle Jim began again, just as we came in sight of the barber’s. I only hope

he said. *Carr* will be at home,Brown’s so clumsy. And Allen’s hand has been shaky ever since he had that fever.

Carr’s

said Uncle Joe.
*certain* to be in,

I’ll bet you sixpence he

said I.
*isn’t*!

Keep your bets for your betters,

said Uncle Joe. I mean

—he hurried on, seeing by the grin on my face what a slip he’d made—I mean that I can

*prove* it, logically. It isn’t a matter of *chance*.

Prove it

sneered Uncle Jim. *logically!*Fire away, then! I defy you to do it!

For the sake of argument,

Uncle Joe began, let us assume Carr to be

*out*. And let us see what that assumption would lead to. I’m going to do this by Reductio ad Absurdum.

Of course you are!

growled Uncle Jim. Never knew any argument of

*yours* that didn’t end in some absurdity or other!

Unprovoked by your unmanly taunts,

said Uncle Joe in a lofty tone, I proceed. Carr being out, you will grant that, if Allen is

*also* out, *Brown* must be at home?

What’s the good of

said Uncle Jim. *his* being at home?I don’t want

*Brown* to shave me! He’s too clumsy.

Patience is one of those inestimable qualities——

Uncle Joe was beginning; but Uncle Jim cut him off short.

he said. *Argue!*Don’t

*moralise!*

Well, but

Uncle Joe persisted. *do* you grant it?Do you grant me that, if Carr is out, it follows that if Allen is out Brown

*must* be in?

Of course he must,

said Uncle Jim; or there’d be nobody in the shop.

We see, then, that the absence of Carr brings into play a certain Hypothetical, whose protasis is

Allen is out,

and whose apodosis is Brown is in

. And we see that, so long as Carr remains out, this Hypothetical remains in force?

Well, suppose it does. What then?

said Uncle Jim.

You will also grant me that the truth of a Hypothetical—I mean its

*validity* as a logical *sequence*—does not in the least depend on its protasis being actually *true*, nor even on its being *possible*. The Hypothetical, If you were to run from here to London in five minutes you would surprise people,

remains true as a *sequence*, whether you can do it or not.

I

said Uncle Jim.
*ca’n’t* do it,

We have now to consider

*another* Hypothetical. What was that you told me yesterday about Allen?

I told you,

said Uncle Jim, that ever since he had that fever he’s been so nervous about going out alone, he always takes Brown with him.

Just so,

said Uncle Joe. Then the Hypothetical,

if Allen is out Brown is out

is *always* in force, isn’t it?

I suppose so,

said Uncle Jim. (He seemed to be getting a little nervous, himself, now.)

Then if Carr is out, we have

*two* Hypotheticals, if Allen is out Brown is

and *in*If Allen is out Brown is

in force at once. And two *out*,*incompatible* Hypotheticals, mark you! They ca’n’t *possibly* be true together!

said Uncle Jim.
*Ca’n’t* they?

How

said Uncle Joe. *can* they?How

*can* one and the same protasis prove two contradictory apodoses? You grant that the two apodoses, Brown is

and *in*Brown is

*out*,*are* contradictory, I suppose?

Yes, I grant

said Uncle Jim.
*that*,

Then I may sum up,

said Uncle Joe. If Carr is out, these two Hypotheticals are true together. And we know that they

*cannot* be true together. Which is absurd. Therefore Carr *cannot* be out. There’s a nice Reductio ad Absurdum for you!

Uncle Jim looked thoroughly puzzled: but after a bit he plucked up courage, and began again. I don’t feel at all clear about that

*incompatibility*. Why shouldn’t those two Hypotheticals be true together? It seems clear to me that would simply prove

. Of course it’s clear that the apodoses of those two Hypotheticals are incompatible—*Allen* is inBrown is in

and Brown is out

. But why shouldn’t we put it like this? If Allen is out Brown is *out*. If Carr and Allen are *both* out, Brown is *in*. Which is absurd. Therefore Carr and Allen ca’n’t be *both* of them out. But, so long as Allen is *in*, I don’t see what’s to hinder Carr from going *out*.

My dear, but most illogical, brother!

said Uncle Joe. (Whenever Uncle Joe begins to dear

you, you may make pretty sure he’s got you in a cleft stick!) Don’t you see that you are wrongly dividing the protasis and the apodosis of the Hypothetical? Its protasis is simply

Carr is out

; and its apodosis is a sort of sub-Hypothetical, If Allen is out, Brown is

. And a most absurd apodosis it is, being hopelessly incompatible with that other Hypothetical that we know is *in**always* true, If Allen is out, Brown is

. And it’s simply the assumption *out*Carr is out

that has caused this absurdity. So there’s only *one* possible conclusion. *Carr is in!*

How long this argument *might* have lasted, I haven’t the least idea. I believe *either* of them could argue for six hours at a stretch. But, just at this moment, we arrived at the barber’s shop; and, on going inside, we found——

The paradox, of which the forgoing paper is an ornamental presentation, is, I have reason to believe, a very real difficulty in the Theory of Hypotheticals. The disputed point has been for some time under discussion by several practised logicians, to whom I have submitted it; and the various and conflicting opinions, which my correspondence with them has elicited, convince me that the subject needs further consideration, in order that logical teachers and writers may come to some agreement as to what Hypotheticals *are*, and how they ought to be treated.

The original dispute, which arose, more than a year ago, between two students of Logic, may be symbolically represented as follows:—

There are two Propositions, `A` and `B`.

It is given that

- (1) If
`C`is true, then, if`A`is true,`B`is not true; - (2) If
`A`is true,`B`is true.

The question is, can `C` be true?

The reader will see that if, in these two Propositions, we replace the letters `A`, `B`, `C` by the names Allen, Brown, Carr, and the words true

and not true

by the words out

and in

we get

- (1) If Carr is out, then, if Allen is out, Brown is in;
- (2) If Allen is out, Brown is out.

These are the very two Propositions on which Uncle Joe

builds his argument.

Several very interesting questions suggest themselves in connexion with this point, such as

Can a Hypothetical, whose protasis is false, be regarded as legitimate?

Are two Hypotheticals, of the forms If

and `A` then `B`If

compatible?
`A` then not-`B`,

What difference in meaning, if any, exists between the following Propositions?

- (1)
`A`,`B`,`C`, cannot be all true at once; - (2) If
`C`and`A`are true,`B`is not true; - (3) If
`C`is true, then, if`A`is true,`B`is not true; - (4) If
`A`is true, then, if`C`is true,`B`is not true.

The following concrete form of the paradox has just been sent me, and may perhaps, as embodying *necessary* truth, throw fresh light on the question.

Let there be three lines, `K``L`, `L``M`, `M``N`, forming, at `L` and `M`, equal acute angles on the same side of `L``M`.

Let

mean `A`The points

.
`K` and `N` coincide, so that the three lines form a triangle

Let

mean `B`The triangle has equal base-angles

.

Let

mean `C`The lines

.
`K``L` and `M``N` are unequal

Then we have

- (1) If
`C`is true, then, if`A`is true,`B`is not true. - (2) If
`A`is true,`B`is true.

The second of these Propositions needs no proof; and the first is proved in Euc., i, 6, though of course it may be questioned whether it fairly represents Euclid’s meaning.

I greatly hope that some of the readers of Mind who take an interest in logic will assist in clearing up these curious difficulties.